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1. Model-Based Multi-Agent RL in Zero-Sum Markov Games with Near-Optimal Sample Complexity

   Zhang, Kaiqing; Kakade, Sham; Basar, Tamer; Yang, Lin (January 2020, Advances in neural information processing systems)

   null (Ed.)

   Model-based reinforcement learning (RL), which finds an optimal policy using an empirical model, has long been recognized as one of the cornerstones of RL. It is especially suitable for multi-agent RL (MARL), as it naturally decouples the learning and the planning phases, and avoids the non-stationarity problem when all agents are improving their policies simultaneously using samples. Though intuitive and widely-used, the sample complexity of model-based MARL algorithms has been investigated relatively much less often. In this paper, we aim to address the fundamental open question about the sample complexity of model-based MARL. We study arguably the most basic MARL setting: two-player discounted zero-sum Markov games, given only access to a generative model of state transition. We show that model-based MARL achieves a near optimal sample complexity for finding the Nash equilibrium (NE) \emph{value} up to some additive error. We also show that this method is near-minimax optimal with a tight dependence on the horizon and the number of states. Our results justify the efficiency of this simple model-based approach in the multi-agent RL setting. 

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2. Near-Optimal Offline Reinforcement Learning via Double Variance Reduction

   Yin, Ming; Bai, Yu; Wang, Yu-Xiang (December 2021, Advances in neural information processing systems)

   We consider the problem of offline reinforcement learning (RL) -- a well-motivated setting of RL that aims at policy optimization using only historical data. Despite its wide applicability, theoretical understandings of offline RL, such as its optimal sample complexity, remain largely open even in basic settings such as \emph{tabular} Markov Decision Processes (MDPs). In this paper, we propose Off-Policy Double Variance Reduction (OPDVR), a new variance reduction based algorithm for offline RL. Our main result shows that OPDVR provably identifies an ϵ-optimal policy with O˜(H2/dmϵ2) episodes of offline data in the finite-horizon stationary transition setting, where H is the horizon length and dm is the minimal marginal state-action distribution induced by the behavior policy. This improves over the best known upper bound by a factor of H. Moreover, we establish an information-theoretic lower bound of Ω(H2/dmϵ2) which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rate-optimal sample complexity under alternative settings such as the finite-horizon MDPs with non-stationary transitions and the infinite horizon MDPs with discounted rewards. 

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3. What are the Statistical Limits of Offline RL with Linear Function Approximation?

   Wang, Ruosong; Foster, Dean; Kakade, Sham M. (January 2021, International Conference on Learning Representations)

   Offline reinforcement learning seeks to utilize offline (observational) data to guide the learning of (causal) sequential decision making strategies. The hope is that offline reinforcement learning coupled with function approximation methods (to deal with the curse of dimensionality) can provide a means to help alleviate the excessive sample complexity burden in modern sequential decision making problems. However, the extent to which this broader approach can be effective is not well understood, where the literature largely consists of sufficient conditions. This work focuses on the basic question of what are necessary representational and distributional conditions that permit provable sample-efficient offline reinforcement learning. Perhaps surprisingly, our main result shows that even if: i) we have realizability in that the true value function of \emph{every} policy is linear in a given set of features and 2) our off-policy data has good coverage over all features (under a strong spectral condition), any algorithm still (information-theoretically) requires a number of offline samples that is exponential in the problem horizon to non-trivially estimate the value of \emph{any} given policy. Our results highlight that sample-efficient offline policy evaluation is not possible unless significantly stronger conditions hold; such conditions include either having low distribution shift (where the offline data distribution is close to the distribution of the policy to be evaluated) or significantly stronger representational conditions (beyond realizability). 

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4. Towards instance-optimal offline reinforcement learning with pessimism

   Yin, Ming; Wang, Yu-Xiang (December 2021, Advances in neural information processing systems)

   We study the \emph{offline reinforcement learning} (offline RL) problem, where the goal is to learn a reward-maximizing policy in an unknown \emph{Markov Decision Process} (MDP) using the data coming from a policy $$\mu$$. In particular, we consider the sample complexity problems of offline RL for the finite horizon MDPs. Prior works derive the information-theoretical lower bounds based on different data-coverage assumptions and their upper bounds are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the \emph{Adaptive Pessimistic Value Iteration} (APVI) algorithm and derive the suboptimality upper bound that nearly matches $$ O\left(\sum_{h=1}^H\sum_{s_h,a_h}d^{\pi^\star}_h(s_h,a_h)\sqrt{\frac{\mathrm{Var}_{P_{s_h,a_h}}{(V^\star_{h+1}+r_h)}}{d^\mu_h(s_h,a_h)}}\sqrt{\frac{1}{n}}\right). $$ We also prove an information-theoretical lower bound to show this quantity is required under the weak assumption that $$d^\mu_h(s_h,a_h)>0$$ if $$d^{\pi^\star}_h(s_h,a_h)>0$$. Here $$\pi^\star$$ is a optimal policy, $$\mu$$ is the behavior policy and $$d(s_h,a_h)$$ is the marginal state-action probability. We call this adaptive bound the \emph{intrinsic offline reinforcement learning bound} since it directly implies all the existing optimal results: minimax rate under uniform data-coverage assumption, horizon-free setting, single policy concentrability, and the tight problem-dependent results. Later, we extend the result to the \emph{assumption-free} regime (where we make no assumption on $$ \mu$$) and obtain the assumption-free intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL. 

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5. Towards optimal off-policy evaluation for reinforcement learning with marginalized importance sampling.

   Xie, T.; Ma, Y.; Wang, Y. X. (January 2019, Advances in neural information processing systems)

   Motivated by the many real-world applications of reinforcement learning (RL) that require safe-policy iterations, we consider the problem of off-policy evaluation (OPE) — the problem of evaluating a new policy using the historical data ob- tained by different behavior policies — under the model of nonstationary episodic Markov Decision Processes (MDP) with a long horizon and a large action space. Existing importance sampling (IS) methods often suffer from large variance that depends exponentially on the RL horizon H. To solve this problem, we consider a marginalized importance sampling (MIS) estimator that recursively estimates the state marginal distribution for the target policy at every step. MIS achieves a mean-squared error of [ ] where μ and π are the logging and target policies, dμt (st) and dπt (st) are the marginal distribution of the state at tth step, H is the horizon, n is the sample size and V π is the value function of the MDP under π. The result matches the t+1 Cramer-Rao lower bound in Jiang and Li [2016] up to a multiplicative factor of H. To the best of our knowledge, this is the first OPE estimation error bound with a polynomial dependence on H . Besides theory, we show empirical superiority of our method in time-varying, partially observable, and long-horizon RL environments. 

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